Fig. REF  depicts the ratio between
the best-fitting cluster distances based on the star count method against the mean cluster distances that we derive from the other methods. We split the cluster sample into two groups, clusters with 10 or more
distance determinations from the other methods and clusters with less than 10 measurements. We expect that the distances of clusters in the first group are well determined so that
any deviation is mainly due to errors in the star count distances. It can be seen that the average distance ratio is well below unity. The reason is probably that the \(N\) -body models that we use to fit
the clusters do not contain primordial binaries. Since the average mass of binaries is larger than that of single stars, real globular clusters contain fewer stellar systems than models with only single stars
for the same total mass, leading to an underprediction of the cluster distances with our method. The size of this underprediction is also in rough agreement with the observed binary fractions, which are around
10% [1]}, [2]}, [3]}. Fig. REF  also indicates that the star count distances contain additional errors that are not accounted for by the uncertainties in the
cluster kinematics, since even when applying a constant shift, the deviations between the star count distances and the mean distances of the other methods are larger than what can be accounted for by the errors in either method.
These additional errors could arise due to for example mismatches between the chosen mass functions of the \(N\) -body models and those of the real clusters or errors in the conversion from
luminosity to mass from the isochrones. We will fit for an additional shift in the star count distances as well as additional errors in sec. 4 when we compare all distances.
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